Integral of 2(1-x) dx

1. The integral of a constant times a function is the constant times the integral of the function:

   $$\int 2 \cdot \left(1 - x\right)\, dx = 2 \int \left(1 - x\right)\, dx$$

   1. Integrate term-by-term:

      1. The integral of a constant is the constant times the variable of integration:

         $$\int 1\, dx = x$$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \left(- x\right)\, dx = - \int x\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x\, dx = \frac{x^{2}}{2}$$

         So, the result is: $- \frac{x^{2}}{2}$

      The result is: $- \frac{x^{2}}{2} + x$

   So, the result is: $- x^{2} + 2 x$

2. Now simplify:

   $$x \left(2 - x\right)$$

3. Add the constant of integration:

   $$x \left(2 - x\right)+ \mathrm{constant}$$

The answer is:

$$x \left(2 - x\right)+ \mathrm{constant}$$